VIPSolutionsLotka-Volterra model Let z;(t), i = 1,2 be populations of two species. Each grows with intrinsic rate r; when resources are infinite: zi =riz, 1;>0 With limited resources, the growth rate becomes r; (1 — 2), where K; > 0. The logistic equation is: 2 T= ra; (1 - 7) 1) Competition between species causes a decrease proportional to the competitor popu- lation: —ayz12, for z1, and —aszezy for zp, with ay, ap > 0. The resulting system: 2 =rz (1-2) — zz 1 1 K, 1.0102 f= PL P) Ty =T2T2 ( Xx, QT2T1 2) Find nullclines of system (2), i.e., lines where z} = 0 or z, = 0. Determine biologi- cally relevant equilibrium points (intersections of nullclines). Show it mathematically.
Question:
Lotka-Volterra model
Let z;(t), i = 1,2 be populations of two species. Each grows with intrinsic rate r; when
resources are infinite:
zi =riz, 1;>0
With limited resources, the growth rate becomes r; (1 — 2), where K; > 0. The
logistic equation is:
2
T= ra; (1 - 7) 1)
Competition between species causes a decrease proportional to the competitor popu-
lation: —ayz12, for z1, and —aszezy for zp, with ay, ap > 0. The resulting system:
2 =rz (1-2) — zz
1 1 K, 1.0102
f= PL P)
Ty =T2T2 ( Xx, QT2T1 2)
Find nullclines of system (2), i.e., lines where z} = 0 or z, = 0. Determine biologi-
cally relevant equilibrium points (intersections of nullclines).
Show it mathematically.
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Asked by: Test User 100193
Created at: 2025-05-07 13:24:28
Asked by: Test User 100193
Created at: 2025-05-07 13:24:28
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Let's **find the nullclines** and **equilibrium points** for the system (2):
### System (2):
\[
\begin{cases}
x_1' = r_1 x_1 \left(1 - \frac{x_1}{K_1}\right) - \alpha_1 x_1 x_2 \\
x_2' = r_2 x_2 \left(1 - \frac{x_2}{K_2}\right) - \alpha_2 x_2 x_1
\end{cases}
\]
---
## **1. Nullclines**
### **\( x_1' = 0 \) nullcline:**
\[
r_1 x_1 \left(1 - \frac{x_1}{K_1}\right) - \alpha_1 x_1 x_2 = 0
\]
Factor \(x_1\):
\[
x_1 \left[ r_1 \left(1 - \frac{x_1}{K_1}\right) - \alpha_1 x_2 \right] = 0
\]
So, either:
- \(x_1 = 0\) **(vertical axis)**
- or
\[
r_1 \left(1 - \frac{x_1}{K_1}\right) - \alpha_1 x_2 = 0
\]
\[
r_1 - r_1 \frac{x_1}{K_1} - \alpha_1 x_2 = 0
\]
\[
r_1 - \alpha_1 x_2 = r_1 \frac{x_1}{K_1}
\]
\[
x_1 = K_1 \left(1 - \frac{\alpha_1}{r_1} x_2\right)
\]
---
### **\( x_2' = 0 \) nullcline:**
\[
r_2 x_2 \left(1 - \frac{x_2}{K_2}\right) - \alpha_2 x_2 x_1 = 0
\]
Factor \(x_2\):
\[
x_2 \left[ r_2 \left(1 - \frac{x_2}{K_2}\right) - \alpha_2 x_1 \right] = 0
\]
So, either:
- \(x_2 = 0\) **(horizontal axis)**
- or
\[
r_2 \left(1 - \frac{x_2}{K_2}\right) - \alpha_2 x_1 = 0
\]
\[
x_2 = K_2 \left(1 - \frac{\alpha_2}{r_2} x_1\right)
\]
---
## **2. Equilibrium Points (Intersections of Nullclines)**
Equilibrium points are where both \(x_1'\) and \(x_2'\) are zero simultaneously.
**Possible cases:**
### **Case 1: \(x_1 = 0, x_2 = 0\)**
- **Origin:** \((0, 0)\)
---
### **Case 2: \(x_1 = 0\), \(x_2 \neq 0\)**
Plug \(x_1 = 0\) into \(x_2'\):
\[
x_2' = r_2 x_2 \left(1 - \frac{x_2}{K_2}\right)
\]
So, \(x_2 = 0\) or \(x_2 = K_2\)
So, equilibrium at \((0, K_2)\)
---
### **Case 3: \(x_2 = 0\), \(x_1 \neq 0\)**
Plug \(x_2 = 0\) into \(x_1'\):
\[
x_1' = r_1 x_1 \left(1 - \frac{x_1}{K_1}\right)
\]
So, \(x_1 = 0\) or \(x_1 = K_1\)
So, equilibrium at \((K_1, 0)\)
---
### **Case 4: Both nonzero, on nontrivial nullclines**
Set:
\[
x_1 = K_1 \left(1 - \frac{\alpha_1}{r_1} x_2\right)
\]
\[
x_2 = K_2 \left(1 - \frac{\alpha_2}{r_2} x_1\right)
\]
Substitute \(x_1\) into the equation for \(x_2\):
\[
x_2 = K_2 \left[1 - \frac{\alpha_2}{r_2} K_1 \left(1 - \frac{\alpha_1}{r_1} x_2 \right)\right]
\]
Expand:
\[
x_2 = K_2 \left[1 - \frac{\alpha_2 K_1}{r_2} + \frac{\alpha_2 K_1 \alpha_1}{r_2 r_1} x_2 \right]
\]
\[
x_2 = K_2 \left(1 - \frac{\alpha_2 K_1}{r_2}\right) + K_2 \frac{\alpha_2 K_1 \alpha_1}{r_2 r_1} x_2
\]
Bring terms with \(x_2\) to one side:
\[
x_2 - K_2 \frac{\alpha_2 K_1 \alpha_1}{r_2 r_1} x_2 = K_2 \left(1 - \frac{\alpha_2 K_1}{r_2}\right)
\]
\[
x_2 \left[1 - K_2 \frac{\alpha_2 K_1 \alpha_1}{r_2 r_1}\right] = K_2 \left(1 - \frac{\alpha_2 K_1}{r_2}\right)
\]
\[
x_2 = \frac{K_2 \left(1 - \frac{\alpha_2 K_1}{r_2}\right)}{1 - K_2 \frac{\alpha_2 K_1 \alpha_1}{r_2 r_1}}
\]
Similarly, you can substitute back to find \(x_1\):
\[
x_1 = K_1 \left(1 - \frac{\alpha_1}{r_1} x_2\right)
\]
---
## **Biologically Relevant Equilibrium Points**
- \((0, 0)\): extinction of both species
- \((K_1, 0)\): species 1 at carrying capacity, species 2 extinct
- \((0, K_2)\): species 2 at carrying capacity, species 1 extinct
- \((x_1^*, x_2^*)\): coexistence equilibrium, given by above formulas (provided both \(x_1^*, x_2^* > 0\))
---
### **Summary Table**
| Equilibrium Point | Meaning |
|--------------------------|------------------------------------------------|
| \((0,0)\) | Both species extinct |
| \((K_1,0)\) | Only species 1 survives |
| \((0,K_2)\) | Only species 2 survives |
| \((x_1^*, x_2^*)\) | Coexistence (if both positive) |
Where
\[
x_2^* = \frac{K_2 \left(1 - \frac{\alpha_2 K_1}{r_2}\right)}{1 - K_2 \frac{\alpha_2 K_1 \alpha_1}{r_2 r_1}}
\]
\[
x_1^* = K_1 \left(1 - \frac{\alpha_1}{r_1} x_2^*\right)
\]
---
**These are the nullclines and equilibrium points mathematically.**
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